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Guidance

Functions symmetrical across the line
(the
axis) are called even. Even functions have the property that when a negative value is substituted for
, it produces the same value as when the positive value is substituted for the
.

Functions that have rotational symmetry about the origin are called odd functions. When a negative
value is substituted into the function, it produces a negative version of the function evaluated at a positive value.

This property becomes increasingly important in problems and proofs of Calculus and beyond, but for now it is sufficient to identify functions that are even, odd or neither and show why.

Example A

Show that
is even.

Solution:

The property that both positive and negative numbers raised to an even power are always positive is the reason why the term even is used. It does not matter that the coefficients are even or odd, just the exponents.

Example B

Show that
is odd.

Solution:

Just like even functions are named, odd functions are named because negative signs don’t disappear and can always be factored out of odd functions.

Example C

Identify whether the function is even, odd or neither and explain why.

Solution:

This does not seem to match either
or
. Therefore, this function is neither even nor odd.

Note that this function is a difference of an odd function and an even function. This should be a clue that the resulting function is neither even nor odd.

Concept Problem Revisited

Even and odd functions describe different types of symmetry, but both derive their name from the properties of exponents. A negative number raised to an even number will always be positive. A negative number raised to an odd number will always be negative.

Vocabulary

An
even
function
means
. Even functions have
reflection symmetry
across the line
.

An
odd function
means
. Odd functions have
rotation symmetry
about the origin.

Guided Practice

1. Which of the basic functions are even, which are odd and which are neither?

2. Suppose
is an even function and
is an odd function.
Is
even or odd?

3. Determine whether the following function is even, odd, or neither.

Answers:

1. Even Functions: The squaring function, the absolute value function.